On the Basic Properties and the Structure of Power Cells

Given a set $$T\subseteq {\mathbb {R}}^{n}$$ T ⊆ R n and a nonnegative function r defined on T, we consider the power of $$x\in {\mathbb {R}}^{n}$$ x ∈ R n with respect to the sphere with center $$t\in T$$ t ∈ T and radius $$r\left( t\right) ,$$ r t , that is, $$ {p_r\left( x,t\right) }:=\left\| x-t\right\| ^{2}-r^{2}\left( t\right) ,$$ p r x , t : = x - t 2 - r 2 t , with $$\left\| \cdot \right\| $$ · denoting the Euclidean distance. The corresponding power cell of $$s\in T$$ s ∈ T is the set $$\begin{aligned} C_{T}^{r}(s):=\{x\in {\mathbb {R}}^{n}:{ p_r}(x,s)\le {p_r}(x,t),\ \text{ for } \text{ all }\ t\in T\}. \end{aligned}$$ C T r ( s ) : = { x ∈ R n : p r ( x , s ) ≤ p r ( x , t ) , for all t ∈ T } . We study the structure of such cells and investigate the assumptions on r that allow for generalizing known results on classical Voronoi cells.